Applied Math Courses

Please see the Math Department’s Course page for this semester’s undergraduate courses.

The following advanced courses in Applied Mathematics will be offered in the Fall of 2004 (please see also the relevant portion of the Graduate Student Handbook):

MA598U : Mixed Least-Squared Finite Element Methods
The original physical equations for mechanics of continua are first-order partial differential systems. There are many advantages to simulate these first-order systems directly. This can be done through either mixed or least-squares finite element methods. This course is an introduction to both techniques, with applications to Darcys flow in porous media, elastic equations for solids, incompressible Newtonian fluid flow, and Maxwells equations in electromagnetic. We shall focus on fundamental issues such as (mixed and least-squares) variational formulations and finite element approximations of important function spaces H1, H(div), and H(curl). A review of fast iterative solvers such as multigrid and domain decomposition for algebraic systems resulting from discretization will also be presented.

MA642 : Methods Of Linear And Nonlinear Partial Differential Equations I
Second order elliptic equations including maximum principles, Harnack inequality, Schauder estimates and Sobolev estimates. Applications of linear theory to nonlinear equations.

MA692B : Special Topics On Navier-Stokes Equations:Numerical Analsis And Implementation

  1. Basic existence and uniqueness theory for the Stokes and Navier-Stokes equations
  2. Numerical Analysis: - Approximations of Stokes equations inf-sup conditions - Approximations of time dependent Navier-Stokes equations - penalty methods, artificial compressibility methods, projection methods
  3. Implementation: - A basic spectral-Galerkin code will be provided for the Navier-Stokes equations. The students will be asked to perform simulations of some model problems such as driven cavity and spin-up.
  4. Other related equations: - Phase-field equations: Allen-Cahn and Cahn-Hillard Coupling of Navier-Stokes equations with phase-field equations

MA694C : Topics On Backward And Forward-Backward Stochastic Differential Equations
This course is a continuation of the MA694A, Introduction to Backward Stochastic Differential Equations offered in Fall 2003, but it will be delivered in a self-contained manner. The preliminaries of BSDEs will be briefly reviewed, and the topics are then expanded to the non-standard cases including forward-backward differential equations, and those with reflections or functional terminals. A thorough study of path-regularity of the solutions and its relation with numerical methods for BSDEs/FBSDEs will be another main topic. If time permits, the applications of BSDEs and FBSDEs to mathematical finance theory will be revisited and further explored.

CS514 : Numerical Analysis
Iterative methods for solving nonlinear equations; linear difference equations, applications to solution of polynomial equations; differentiation and integration formulas; numerical solution of ordinary differential equations; roundoff error bounds.